We present a numerical study of a two-lane version of the stochastic nonequilibrium model known as the totally asymmetric simple exclusion process. For such a system with open boundaries, and suitably chosen values of externally imposed particle injection (α) and ejection (β) rates, spontaneous symmetry breaking can occur. We investigate the statistics and internal structure of the stochastically induced transitions or "flips," which occur between opposite broken-symmetry states as the system evolves in time. From the distribution of time intervals separating successive flips, we show that the evolution of the associated characteristic times against externally imposed rates yields information regarding the proximity to a critical point in parameter space. On short timescales, we probe for the possible existence of precursor events to a flip between opposite broken-symmetry states. We study an adaptation of domain-wall theory to mimic the density reversal process associated with a flip.

We present a simplified description for spin-dependent electronic transport in honeycomb-lattice structures with spin-orbit interactions, using generalizations of the stochastic non-equilibrium model known as the totally asymmetric simple exclusion process. Mean field theory and numerical simulations are used to study currents, density profiles and current polarization in quasione dimensional systems with open boundaries, and externally-imposed particle injection (α) and ejection (β) rates. We investigate the influence of allowing for double site occupancy, according to Pauli’s exclusion principle, on the behavior of the quantities of interest. We find that double occupancy shows strong signatures for specific combinations of rates, namely high α and low β, but otherwise its effects are quantitatively suppressed. Comments are made on the possible relevance of the present results to experiments on suitably doped graphenelike structures.

We have studied a system of stochastically reorienting leftand right-pointing arrowheads ( < and > ) in one dimension. A typical configuration can be ....<<>>><>><>>><>>><<>>>>>.... The kinetics involves lateral flipping of an arrowhead around its vertex and displacement along the line. The arrowheads can not overlap or cross each other due to hardcore interaction between them. Our study includes the static properties and the dynamics of approach to the steady state. The equilibrium state of our system approaches an orientationally ordered state as the density increases. The spatial extent of this order is quantified by the correlation length. Using transfer matrix formalism it is shown that the correlation length diverges exponentially with increase in arrowhead density. It can be seen from a typical configuration above that there are two types of domain walls, >< (A) and <> (B). In the very high density limit, the displacement of an arrowhead is ineffective so kinetics is reduced to move of flipping only. Thus, owing to non-overlap constraint due to hardcore interaction, A-type domain walls can diffuse and B-type walls are static. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain wall annihilation A + B → 0, quite different from A+A→ 0 kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is analytically shown to decay exponentially in time, in contrast to the power law decay known for A+A→ 0. In the thermodynamic limit with a finite density of walls, coarsening as a function of time is studied by simulation. It is observed that the number of domain walls falls as t− 1 2 . The fraction of persistent arrowheads is found to decay as a power law t−θ where θ is close to 1/4, quite different from the exact value 3/8 for the Ising case. The global persistence also decays with same power θ=1/4. In a generalization where the B-type walls can diffuse slowly, θ varies continuously, increasing with increasing diffusion constant.

We demonstrate the large-scale effects of the interplay between shape and hard-core interactions in a system with left- and right-pointing arrowheads <> on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain walls, >< (A) and <> (B). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In this high-density limit, the A domain walls diffuse, while the B walls are static. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain-wall annihilation A+B→0, quite different from the A+A→0 kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is shown to decay exponentially with time, in contrast to the power-law decay known for A+A→0. In the thermodynamic limit with a finite density of walls, coarsening as a function of time t is studied by simulation. While the number of walls falls as t^{-1/2}, the fraction of persistent arrowheads decays as t^{-θ} where θ is close to 1/4, quite different from the Ising value. The global persistence too has θ=1/4, as follows from a heuristic argument. In a generalization where the B walls diffuse slowly, θ varies continuously, increasing with increasing diffusion constant.

We study the dynamical evolution toward steady state of the stochastic nonequilibrium model known as the totally asymmetric simple exclusion process, in both uniform and nonuniform (staggered) one-dimensional systems with open boundaries. Domain-wall theory and numerical simulations are used and, where pertinent, their results are compared to existing mean-field predictions and exact solutions where available. For uniform chains we find that the inclusion of fluctuations inherent to the domain-wall formulation plays a crucial role in providing good agreement with simulations, which is severely lacking in the corresponding mean-field predictions. For alternating-bond chains the domain-wall predictions for the features of the phase diagram in the parameter space of injection and ejection rates turn out to be realized only in an incipient and quantitatively approximate way. Nevertheless, significant quantitative agreement can be found between several additional domain-wall theory predictions and numerics.

We demonstrate the large scale effects of the interplay between shape and hard core interactions in a system with left- and right-pointing arrowheads ~$\textless ~~ \textgreater$~ on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain wall, diffusive ($A$) and static ($B$). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain wall annihilation $A+B\rightarrow 0$, quite different from $A+A \rightarrow 0$ kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is shown to decay exponentially in time, in contrast to the power law decay known for $A+A \rightarrow 0$. In the thermodynamic limit with a finite density of walls, coarsening as a function of time $t$ is studied by simulation. While the number of walls falls as $t^{-\frac{1}{2}}$, the fraction of persistent arrowheads decays as $t^{-\theta}$ where $\theta$ is close to $\frac{1}{4}$, quite different from the Ising value. The global persistence too has $\theta=\frac{1}{4}$, as follows from a heuristic argument. In a generalization where the $B$ walls diffuse slowly, $\theta$ varies continuously, increasing with increasing diffusion constant.

We present a mean-field theory for the dynamics of driven flow with exclusion in graphenelike structures, and numerically check its predictions. We treat first a specific combination of bond transmissivity rates, where mean field predicts, and numerics to a large extent confirms, that the sublattice structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in the various regions of the phase diagram set by open boundary injection and ejection rates, is then in general identical to that of one-dimensional systems, although some discrepancies remain between mean-field theory and numerical results, in similar ways for both geometries. However, at the critical point for which the characteristic exponent is z = 3/2 in one dimension, the mean-field value z = 2 is approached for very large systems with constant (finite) aspect ratio. We also treat a second combination of bond (and boundary) rates where, more typically, sublattice distinction persists. For the two rate combinations, in continuum or late-time limits, respectively, the coupled sets of mean-field dynamical equations become tractable with various techniques and give a two-band spectrum, gapless in the critical phase. While for the second rate combination quantitative discrepancies between mean-field theory and simulations increase for most properties and boundary rates investigated, theory still is qualitatively correct in general, and gives a fairly good quantitative account of features such as the late-time evolution of density profile differences from their steady-state values.

We collect results for bond percolation on various lattices from two to fourteen dimensions that, in the limit of large dimension d or number of neighbors z, smoothly approach a randomly diluted Erdős-Rényi graph. We include results on bond-diluted hypersphere packs in up to nine dimensions, which show the mean coordination, excess kurtosis, and skewness evolving smoothly with dimension towards the Erdős-Rényi limit.

We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination 〈 r 〉=4.41, which is above the ‘mean field’ value 〈 r 〉=4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is β =0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν =3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination 〈 r 〉=3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.

We consider the one-dimensional Glauber dynamics with coupling disorder in terms of bilinear fermion Hamiltonians. Dynamic exponents embodied in the spectrum gap of these latter are evaluated numerically by averaging over both binary and Gaussian disorder realizations. In the first case, these exponents are found to follow the nonuniversal values of those of plain dimerized chains. In the second situation their values are still nonuniversal and subdiffusive below a critical variance above which, however, the relaxation time is suggested to grow as a stretched exponential of the equilibrium correlation length.

We consider steady-state current activity statistics for the one-dimensional totally asymmetric simple exclusion process. With the help of the known operator algebra (for general open boundary conditions), as well as general probabilistic concepts (for the periodic case), we derive and evaluate closed-form expressions for the lowest three moments of the probability distribution function. These are confirmed, to excellent degree of accuracy, by numerical simulations. Further exact expressions and asymptotic approximations are provided for probability distributions and generating functions.

Renormalization group calculations are used to give exact solutions for rigidity percolation on hierarchical lattices. Algebraic scaling transformations for a simple example in two dimensions produce a transition of second order, with an unstable critical point and associated scaling laws. Values are provided for the order parameter exponent $\beta = 0.0775$ associated with the spanning rigid cluster and also for $d \nu = 3.533$ which is associated with an anomalous lattice dimension $d$ and the divergence in the correlation length near the transition. In addition we argue that the number of floppy modes $F$ plays the role of a free energy and hence find the exponent $\alpha$ and establish hyperscaling. The exact analytical procedures demonstrated on the chosen example readily generalize to wider classes of hierarchical lattice.

We consider the one-dimensional totally asymmetric simple exclusion process (TASEP) with position-dependent hopping rates. The problem is solved, in a mean-field adiabatic approximation, for a general (smooth) form of spatial rate variation. Numerical simulations of systems with hopping rates varying linearly against position (constant rate gradient), for both periodic and open-boundary conditions, provide detailed confirmation of theoretical predictions, concerning steady-state average density profiles and currents, as well as open-system phase boundaries, to excellent numerical accuracy.

Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in the presence of a confining magnetic field which varies with distance |x| to a "trap center," proportionally to (|x|/ℓ)p, where p>0 . On a strip geometry, the competition between the "trap size" ℓ and the strip width L is analyzed in the context of a generalized finite-size scaling ansatz. In the low-field regime ℓ >> L, we use conformal-invariance concepts in conjunction with a linear-response approach to derive the appropriate (p-dependent) limit of the theory, which agrees very well with numerical results for magnetization profiles. For high fields ℓ ≲ L, correlation-length scaling data broadly confirm an existing picture of p-dependent characteristic exponents. Standard spin-1/2 and spin-1 Ising systems are considered, as well as the Blume-Capel model.

We investigate finite-size scaling aspects of disorder reaction-diffusion processes in one dimension utilizing both numerical and analytical approaches. The former averages the spectrum gap of the associated evolution operators by doubling their degrees of freedom, while the latter uses various techniques to map the equations of motion to a first-passage time process. Both approaches are consistent with nonuniversal dynamic exponents and with stretched exponential scaling forms for particular disorder realizations.

We study conserved one-dimensional models of particle diffusion, attachment, and detachment from clusters, where the detachment rates decrease with increasing cluster size as gamma(m) approximately m(-k), k>0 . Heuristic scaling arguments based on random walk properties show that the typical cluster size scales as (t/ln t)z , with z=1/(k+2) . The coarsening of neighboring clusters is characterized by initial symmetric flux of particles between them followed by an effectively asymmetric flux due to the unbalanced detachment rates, which leads to the above logarithmic corrections. Small clusters have densities of order t(-mz)(1), with z(1)=k/(k+2) . Thus for k<1 , the small clusters (mass of order unity) are statistically dominant and the average cluster size does not scale as the size of typically large clusters does. We also solve the master equation of the model under an independent interval approximation, which yields cluster distributions and exponent relations and gives the correct dominant coarsening exponent after suitable changes to incorporate effects of correlations. The coarsening of typical large clusters is described by the distribution Pt(m) approximately 1/ty f(m/tz) , with y=2z . All results are confirmed by simulation, which also illustrates the unusual features of cluster size distributions, with a power-law decay for small masses and a negatively skewed peak in the scaling region. The detachment rates considered here can apply in the presence of strong attractive interactions, and recent applications suggest that even more rapid rate decays are also physically realistic.

We study one-dimensional models of particle diffusion and attachment and/or detachment from islands where the detachment rates gamma of particles at the cluster edges increase with cluster mass m. They are expected to mimic the effects of lattice mismatch with the substrate and/or long-range repulsive interactions that work against the formation of long islands. Short-range attraction is represented by an overall factor epsilon << 1 in the detachment rates (masses m >= 2) relatively to isolated particle hopping rates (epsilon similar to e(-E/T), where E is a binding energy and T is the temperature). We consider various mass dependences of gamma(m) from rapidly increasing forms such as gamma(m)similar to m to slowly increasing ones, such as gamma(m)similar to(m/m+1)(b), with constant b. A mapping onto a column problem shows that these systems are zero-range processes, whose steady state properties are exactly calculated under the assumption of independent column heights in the master equation. Simulation provides accurate island size distributions which confirm analytic reductions and are particularly useful whenever the analytical tools cannot provide results in closed form. The shape of island size distributions can be changed from monomodal to monotonically decreasing by tuning the temperature or changing the coverage (one-dimensional density rho). In all cases, small values of the scaling variable X equivalent to epsilon(-1)rho/(1-rho) favor the monotonically decreasing ones. However, for large X, rapidly increasing rates gamma(m) lead to distributions with peaks very close to < m > and rapidly decreasing tails, while slowly increasing gamma(m) provide peaks close to < m >/2 and fat right tails.

Properties of the one-dimensional totally asymmetric simple exclusion process (TASEP), and their connection with the dynamical scaling of moving interfaces described by a Kardar-Parisi-Zhang equation are investigated. With periodic boundary conditions, scaling of interface widths (the latter defined via a discrete occupation-number-to-height mapping), gives the exponents alpha=0.500(5) , z=1.52(3) , beta=0.33(1) . With open boundaries, results are as follows: (i) in the maximal-current phase, the exponents are the same as for the periodic case, and in agreement with recent Bethe ansatz results; (ii) in the low-density phase, curve collapse can be found to a rather good extent, with alpha=0.497(3) , z=1.20(5) , beta=0.41(2) , which is apparently at variance with the Bethe ansatz prediction z=0 ; (iii) on the coexistence line between low- and high-density phases, alpha=0.99(1) , z=2.10(5) , beta=0.47(2) , in relatively good agreement with the Bethe ansatz prediction z=2 . From a mean-field continuum formulation, a characteristic relaxation time, related to kinematic-wave propagation and having an effective exponent z;{'}=1 , is shown to be the limiting slow process for the low-density phase, which accounts for the above mentioned discrepancy with Bethe ansatz results. For TASEP with quenched bond disorder, interface width scaling gives alpha=1.05(5) , z=1.7(1) , beta=0.62(7) . From a direct analytic approach to steady-state properties of TASEP with quenched disorder, closed-form expressions for the piecewise shape of averaged density profiles are given, as well as rather restrictive bounds on currents. All these are substantiated in numerical simulations.

Mustansir Barma, 2 Marcelo D. Grynberg, and Robin B. Stinchcombe 2 Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India Isaac Newton Institute for Mathematical Sciences, 20, Clarkson Road, Cambridge CB3 0EH, UK Departamento de F́ısica, Universidad Nacional de La Plata, (1900) La Plata, Argentina Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK

To elucidate slow dynamics in glassy materials, we introduce the figure-8 model in which N hard blocks undergo Brownian motion around a circuit in the shape of a figure 8. This system undergoes kinetic arrest at a critical packing fraction phi=phi g<1 , and for phi approximately phi g long-time diffusion is controlled by rare, cooperative, "junction-crossing" particle rearrangements. We find that the average time between junction crossings tau JC, and hence the structural relaxation time, does not simply scale with the configurational volume Omega c of transition states, because tau JC also depends on the time to complete a junction crossing. The importance of these results in understanding cage-breaking dynamics in glassy systems is discussed.